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Theorem imasf1oxmet 22557

Description: The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasf1oxmet.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasf1oxmet.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1oxmet.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxmet.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasf1oxmet.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasf1oxmet.d	⊢ 𝐷 = (dist‘𝑈)
imasf1oxmet.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))

**Assertion**
Ref	Expression
imasf1oxmet	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Proof of Theorem imasf1oxmet Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1oxmet.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasf1oxmet.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1oxmet.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		f1ofo 6389	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
6		imasf1oxmet.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7		eqid 2825	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
8		imasf1oxmet.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
9	1, 2, 5, 6, 7, 8	imasdsfn 16534	. . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
10	1	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “_s 𝑅))
11	2	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
12	3	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵)
13	6	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍)
14		imasf1oxmet.e	. . . . . . . 8 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
15		imasf1oxmet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
16	15	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉))
17		simprl 787	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 789	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19	10, 11, 12, 13, 14, 8, 16, 17, 18	imasdsf1o 22556	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
20		xmetcl 22513	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ^*)
21	20	3expb 1153	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
22	15, 21	sylan 575	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2906	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
24	23	ralrimivva 3180	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
25		f1ofn 6383	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
26	3, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
27		oveq2 6918	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏)))
28	27	eleq1d 2891	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
29	28	ralrn 6616	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
31		forn 6360	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
32	5, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
33	32	raleqdv 3356	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
34	30, 33	bitr3d 273	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
35	34	ralbidv 3195	. . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
36	24, 35	mpbid 224	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*)
37		oveq1 6917	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦))
38	37	eleq1d 2891	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
39	38	ralbidv 3195	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
40	39	ralrn 6616	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
41	26, 40	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
42	32	raleqdv 3356	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
43	41, 42	bitr3d 273	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
44	36, 43	mpbid 224	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*)
45		ffnov 7029	. . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
46	9, 44, 45	sylanbrc 578	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
47		xmeteq0 22520	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
48	16, 17, 18, 47	syl3anc 1494	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
49	19	eqeq1d 2827	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝑎𝐸𝑏) = 0))
50		f1of1 6381	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
51	3, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝐵)
52		f1fveq 6779	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
53	51, 52	sylan 575	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
54	48, 49, 53	3bitr4d 303	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
55	16	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐸 ∈ (∞Met‘𝑉))
56		simpr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
57	17	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑎 ∈ 𝑉)
58	18	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑏 ∈ 𝑉)
59		xmettri2 22522	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
60	55, 56, 57, 58, 59	syl13anc 1495	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
61	19	adantr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
62	10	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑈 = (𝐹 “_s 𝑅))
63	11	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
64	12	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐹:𝑉–1-1-onto→𝐵)
65	13	adantr 474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑅 ∈ 𝑍)
66	62, 63, 64, 65, 14, 8, 55, 56, 57	imasdsf1o 22556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑎)) = (𝑐𝐸𝑎))
67	62, 63, 64, 65, 14, 8, 55, 56, 58	imasdsf1o 22556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑏)) = (𝑐𝐸𝑏))
68	66, 67	oveq12d 6928	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) = ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
69	60, 61, 68	3brtr4d 4907	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
70	69	ralrimiva 3175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
71		oveq1 6917	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑎)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑎)))
72		oveq1 6917	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑏)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))
73	71, 72	oveq12d 6928	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) = (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
74	73	breq2d 4887	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
75	74	ralrn 6616	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
76	26, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
77	32	raleqdv 3356	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
78	76, 77	bitr3d 273	. . . . . . . 8 ⊢ (𝜑 → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
79	78	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
80	70, 79	mpbid 224	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
81	54, 80	jca 507	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
82	81	ralrimivva 3180	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
83	27	eqeq1d 2827	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0))
84		eqeq2 2836	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
85	83, 84	bibi12d 337	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ↔ (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏))))
86		oveq2 6918	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → (𝑧𝐷𝑦) = (𝑧𝐷(𝐹‘𝑏)))
87	86	oveq2d 6926	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
88	27, 87	breq12d 4888	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
89	88	ralbidv 3195	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
90	85, 89	anbi12d 624	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
91	90	ralrn 6616	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
92	26, 91	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
93	32	raleqdv 3356	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
94	92, 93	bitr3d 273	. . . . 5 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
95	94	ralbidv 3195	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
96	82, 95	mpbid 224	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
97	37	eqeq1d 2827	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷𝑦) = 0))
98		eqeq1 2829	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 = 𝑦 ↔ (𝐹‘𝑎) = 𝑦))
99	97, 98	bibi12d 337	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦)))
100		oveq2 6918	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑧𝐷𝑥) = (𝑧𝐷(𝐹‘𝑎)))
101	100	oveq1d 6925	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))
102	37, 101	breq12d 4888	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
103	102	ralbidv 3195	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
104	99, 103	anbi12d 624	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
105	104	ralbidv 3195	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
106	105	ralrn 6616	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
107	26, 106	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
108	32	raleqdv 3356	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
109	107, 108	bitr3d 273	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
110	96, 109	mpbid 224	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
111	15	elfvexd 6472	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
112		fornex 7402	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
113	111, 5, 112	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
114		isxmet 22506	. . 3 ⊢ (𝐵 ∈ V → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
116	46, 110, 115	mpbir2and 704	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 class class class wbr 4875 × cxp 5344 ran crn 5347 ↾ cres 5348 Fn wfn 6122 ⟶wf 6123 –1-1→wf1 6124 –onto→wfo 6125 –1-1-onto→wf1o 6126 ‘cfv 6127 (class class class)co 6910 0cc0 10259 ℝ^*cxr 10397 ≤ cle 10399 +_𝑒 cxad 12237 Basecbs 16229 distcds 16321 “_s cimas 16524 ∞Metcxmet 20098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-0g 16462 df-gsum 16463 df-xrs 16522 df-imas 16528 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-mulg 17902 df-cntz 18107 df-cmn 18555 df-xmet 20106

This theorem is referenced by: imasf1omet 22558 xpsxmet 22562 imasf1obl 22670 imasf1oxms 22671

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